\documentclass[12pt]{article} 

\topmargin=-0.1in 
\textheight=8.5in 

\begin{document}

\noindent{\bf HW \#3, Math 269C } {\bf \ \ \ Due on Monday, May 8}

\noindent{\bf [1]} Consider the problem with an inhomogeneous boundary condition, 
$$
\left\{
\begin{array}{l} 
-\triangle u=f\mbox{ in } \Omega,\\
u=u_0 \mbox{ on } \Gamma=\partial\Omega,
\end{array}
\right.
$$
where $f$ and $u_0$ are given. \underline{Briefly} show that this problem can be given the 
following equivalent formulations: 

$(V)$ Find $u\in V(u_0)$ such that $a(u,v)=(f,v)$, $\forall v\in H^1_0(\Omega)$,

$(M)$ Find $u\in V(u_0)$ such that $F(u)\leq F(v)$, $\forall v\in V(u_0)$,

\noindent where $V(u_0)=\{v\in H^1(\Omega): \ v=u_0 \mbox{ on } \Gamma\}$ and $F$ is defined in the usual way, $F(v)=\frac{1}{2}a(v,v)-(f,v)$. 

Then formulate a finite element method and prove an error estimate (as in Thm. 1.1, page 24). 

Recall: $H_0^1(\Omega)=\{v\in L^2(\Omega), \ \nabla v\in L^2(\Omega)^n,\ v=0\mbox{ on }\partial\Omega\}$, where $n$ is the spatial dimension. 

\bigbreak


\noindent{\bf [2]} (a) Give a weak variational formulation of the problem 
$$\frac{d^4 u}{dx^4}=f\ \mbox{ for }\ 0<x<1,$$
$$u(0)=u''(0)=u'(1)=u'''(1)=0,$$
and show that the assumptions of the Lax-Milgram Lemma are satisfied. Which boundary conditions
are essential and which are natural ? 

(b) Solve the same problem with the following alternative boundary conditions: 
$$u(0)=-u''(0)+\gamma u'(0)=0,\ \ u(1)=u''(1)+\gamma u'(1)=0,$$
where $\gamma$ is a positive constant. 

\bigbreak 

\noindent{\bf [3]}  Give a weak variational formulation of the Robin's problem 
$$-\triangle u=f\ \mbox{ in }\Omega,\ \ \gamma u+\frac{\partial u}{\partial n}=g
\mbox{ on }\Gamma,$$
where $\gamma$ is a constant. When are the assumptions of the Lax-Milgram Lemma satisfied ? 

\bigbreak

\noindent{\bf [4]}  Consider the Neumann problem 
\begin{eqnarray*}
-\triangle u =f \mbox{ in }\Omega,\\
\frac{\partial u}{\partial n}=g\mbox{ on }\Gamma=\partial\Omega,\\
\int_{\Omega}u(x)dx=0.
\end{eqnarray*}
where $f:\Omega\rightarrow R$ and $g:\partial\Omega\rightarrow R$ satisfy the {\it compatibility condition} 
$$\int_{\Omega}f(x)dx+\int_{\partial\Omega}g(x)d\sigma(x)=0.$$

(a) Why condition ``$\int_{\Omega}u(x)dx=0$'' was added here ? Why do we need the compatibility condition ? 

(b) Give a weak variational formulation of the problem,  
and prove that the conditions of the Lax-Milgram Lemma are satisfied, under the necessary assumptions on $f$ and $g$ that you would specify. 

\end{document}
